Optimal interpolants on Grassmann manifolds

Abstract

The Grassmann manifold$Gr_m({\mathbb {R}}^n)$ of all m-dimensional subspaces of the n-dimensional space ${\mathbb {R}}^n$$(m<n)$ is widely used in image analysis, statistics and optimization. Motivated by interpolation in the manifold $Gr_2({\mathbb {R}}^4)$, we first formulate the differential equation for desired interpolation curves called Riemannian cubics in symmetric spaces by the Pontryagin maximum principle (PMP) and then narrow down to it in $Gr_2({\mathbb {R}}^4)$. Although computation on this low-dimensional manifold may not occur heavy burden for modern machines, theoretical analysis for Riemannian cubics is very limited in references due to its highly nonlinearity. This paper focuses on presenting analytical and geometrical structures for the so-called Lie quadratics associated with Riemannian cubics. By analysing asymptotics of Lie quadratics, we find asymptotics of Riemannian cubics in $Gr_2({\mathbb {R}}^4)$. Finally, we illustrate our results by numerical simulations.